Math 131      Topological spaces and the fundamental group

Harvard Fall 2008

Instructor: Thomas Lam

Lectures: Monday, Wednesday, Friday 12-1      Science Center 507
Office Hours: Wednesday 2-3pm.      Science Center 435

Topology, Munkres.

Syllabus: First Day Handout.

Grading: Problem sets (40%), Mid-Term (20%), Exam (40%).

Announcements: (old announcements are moved to bottom of the page)

Final Exam Grading Blog
5:40pm: started grading.
5:58pm: Average on TF questions: 6.7/10. Hardly anyone believed that there was a covering map from the three-fold torus to the two-fold torus. Obviously this question wasn't completely fair; the way to visualize the covering map is to take a 3-holed torus, with the holes arranged in a row. Now rotate the torus around the central hole. The quotient space under this rotation is a 2-holed torus.
7:13pm: Congratulations to the student who correctly calculated (guessed?) the familiar space of Problem 12 a)!
7:17pm: Pretty much decided on grading scheme.
8:53pm: Finally finished grading one exam completely. Perhaps time for dinner?
11:30pm: Getting close to half way. The later questions are rather sparsely done, but I'm quite impressed by the care some of you have shown in tackling some of the subtleties, especially early on..
12:15am: 6 more papers to go...
12:45am: A very common thing tried to provide a counterexample in Problem 9 was to show that f(z) = z^2 and f(z) = z on the circle are homotopic. The homotopy H(z,t) = z^(1+t) looks tempting, but actually isn't well defined. But... nice try!!! :)
1:45am: Done!!!!!!!!!!
2:00am: Statistics: Average is 56. Highscore is 76. Total points awarded for Problem 12 is 1. Every problem was solved by someone, except for Problems 3(c), 9 and 12. Many scored 3 or 4 on Problem 9. For 3(c), many got the answer, but I was not completely convinced by anyone's proof.
2:00am: I'm not allowed to tell you your final grade, but if you email me I can tell you roughly how you did on the final exam.
2:00am: Good work, everybody!

Comments about Final Exam Problems:
The final exam
Problem 2e): You can't cancel bb^{-1}, because there are other occurrences of b in this word.
Problem 3e): Suprisingly few students got this problem. The only homomorphism from Z/5Z to Z is the trivial one, thus the map can be lifted to the covering space R. Since R is contractible, the lift is null-homotopic. Projecting back down shows that the original map is null-homotopic.
Problem 5: This problem is harder than it looks. The argument is "Step 2" of the proof of Theorem 78.1 in the book. (I had, perhaps erroneously, assumed that most of you would have read this proof, or remembered it from the lectures.)
Problem 6: Let U_\alpha be an open cover of E. For each b in B, and each e in p^{-1}(b), pick some U_{b,e} which contains e. By possibly shrinking the U_{b,e}, we assume that U_{b,e} is homeomorphic to p(U_{b,e}) (just intersect with a slice of an even covering). Define V_b = \cap_e p(U_{b,e}). ch is open. Since B is compact, finitely many V_b1, V_b2, ..., V_bn cover B. Then the corresponding U_{bi,e}'s is finite, and will cover E.
Problem 8a): This is NOT a grid Z^2 with edges, which would not be simply connected. The answer is an infinite four-valent tree.
Problem 8c): The covering space of P^2 wedge P^2 is infinitely many 2-spheres glued together in a line. Seems like only two students got this problem. Congratulations!
Problem 9: I suspect no one got this completely (but I need to read the solutions carefully). Many students thought the answer was "True" and gave a proof similar to Lemma 58.1 in the book. The Lemma in the book has an additional requirement that the basepoint is fixed during the homotopy. Here is a counterexample I came up with. Let X be the wedge sum of two S^1's. Start with the identity map from X to X. Now pull the basepoint around one of the loops, and move other points in a way that this is continuous. The resulting map f:X to X is homotopic to the identity map, but it induces conjugation by a loop on \pi_1(X).
Okay let me not be lazy and actually write it in coordinates. Let the two circles be A, B and write A(s) for a point on A, parametrized by s in [0,1]. We have A(0) = A(1) = B(0) = B(1). The function f takes the interval [A(1/4),A(1/2)] to [A(0),A(1/2)]. It takes [A(0),A(1/4)] to [B(0),B(1)]. It takes [A(1/2),A(3/4)] to [A(1/2),A(1)]. It takes [A(3,4),A(1)] to [B(1),B(0)]. It takes B to itself. The homotopy rotates B at constant speed, and also moves every other point to its destination at constant speed.
Problem 10: You were supposed to use Lemma 58.4. In the notation of that Lemma, k_* and h_* are both trivial, so we deduce that conjugation by \alpha = F(x_0,t) is the identity map on \pi_1(X,x_0). This is equivalent to saying that \alpha lies in the center.
Problem 11: This is discussed (without proof) on p.288 of the book. I wasn't actually expecting anyone to have studied this, but the proofs are similar to the point-set topology proofs in Chapter 7.
Problem 12: X is S^2. For b), R^3 - A is the wedge sum of two S^1-s and two S^2-s. R^3-A is the wedge sum of S^1 with a torus. I wasn't actually expecting anyone to figure this out...

Problem sets: There will be problem sets once a week. Collaboration on homework is encouraged, but you are not allowed to just copy someone else's work. You have to mention on your problem set who you worked with.

Pset 1 (Due 09/22): 13.1, 13.3, 13.8, 17.7, 17.8, 17.16, 18.9
Q1: Let Y be a Hausdorff space and f, g:X -> Y be two continuous functions. Prove that {x| f(x) = g(x)} is closed in X.
Q2: Send an email to me ( telling me about your mathematics background; in particular, I'd like to know how familiar you are with set theory (e.g. intersections, unions, complements) and abstract algebra (do you know what a "group" is?)

Pset 2 (Due 09/29): 19.4, 19.7, 20.4, 20.5, 20.8, 21.6, 21.10.
Q1: At the bottom of p.131 of the book, Munkres says "Uniformity of converegence depends not only on the topology of Y but also on its metric". Explain this sentence.

Solution: let X be the one-point topological space, and Y be R^\omega with the product topology. Define f_n:X --> Y by f_n(x) = (0,0,...,0,n,0,...) where the n is in the n-th position. Define f:X -->Y by f_n(x) = (0,0,0,...). Using the metric in Theorem 20.5, these functions f_n do not uniformly converge to f, because D(f_n(x),f(x)) = 1. However, f_n uniformly converge to f if we use the metric D'(x,y) = sup{ d(x_i,y_i)/i^2}, where d is the standard bounded metric on R. Check yourself: D' induces the product topology on R^\omega.

Pset 3 (Due 10/06): 23.4, 23.5, 24.1, 26.2
Q1: Find counterexamples to Theorem 26.6 when X is not compact, or Y is not Hausdorff.
Q2: Do the universal property problems.
Q3: Prove carefully that the map f:[0,2pi]->S^1 given by f(x) = (cosx,sinx) is a quotient map. You may assume that it is known that sinx and cosx are continuous.

John's select solutions to Psets 1-3

Pset 4 (Due 10/15): 26.7, 26.8, 27.3, 30.12, 31.2, 31.7, 32.1, 32.6.

Pset 5 (Due 10/20): Either (read Theorem 32.4 and do Exercise 32.8) or (do Exercise 32.9). Do 43.1, 43.5.

Pset 6 (due 10/27): 45.2(a)(b), 45.7, 46.5, 46.7, 47.3.

Pset 7 (due 11/10): 51.3, 52.3, 52.6, 52.7, 53.3, 53.4. In 53.4, explain, preferably with an example, what could go wrong if r^{-1}(z) is not finite.
Q1: Classify, as carefully as you can, the compact 1-manifolds. (Optional: remove the condition "compact", but you may want to add countable basis in that case).

Pset 8 (due 11/17): 54.4, 54.6, 79.1, 79.3, 79.4, (either 79.5 or 79.6). (If your group theory is rusty, you may want to read the start of Chapter 11.)

Pset 9 (due 11/24): 58.2, 58.6, 58.9, 80.1(a), 81.2(a)(b)
Q1: Skim through sections 67-69 of the book -- I will not spend too much time on free groups and free products in class. Do 69.1.

Pset 10 (due 12/8): 60.4, 60.5, 70.1, 71.1, 71.2, 71.5, 72.2, 73.1.

Pset 10 (due 12/15): 74.1, 74.3, 74.4(a), 75.1.

Information about the Exam:
1) There are 105 points. 100 points is a full score.
2) There are 12 questions. The last one is the "bonus" 5 point question, which I am not expecting anyone to do completely; but it may amuse you if you are bored during the exam.
3) The first question, worth 10 points, is ten TF-questions.
4) The second question, worth 15 points, asks you to figure out the fundamental groups of six (different?) spaces.
5) The third question is worth 18 points, and is a series of five questions, most of which have a yes/no answer which you need to prove or give a counterexample.
6) The fourth question asks you to figure out which surface a polygon with labeled sides is homeomorphic to, using cut-and-paste operations.
7) The questions 5-11 are worth 5-10 points each, and are independent. Most of them are "Prove that..." problems, but some ask for topological spaces, groups, or maps as an answer. The average one is not too different to exercises in the book; in fact one or two may actually be in the book.

Practice problems for final exam:
53.5, 57.3, 58.7, 60.3, 72.1, 73.2, 76.2, 77.3(f), 79,7.
This list is not meant to be comprehensive, nor meant to be an indication of the length or difficulty of the exam.

During the review session of Jan 12, I mentioned that you do not need to know about orbit spaces (p.490-491 of the book).

Information about Final Exam:

Past Exams for practice: 2002 2004

Format: The format of the exam will be similar to the MidTerm, except it will be around twice as long.

Syllabus: The exam will focus on the second half of the course, but problems will still require you to be able to do point-set topology. It is unlikely that I will ask you to solve pure point-set topology problems by themselves, without relations to something taught in the second half. For example, one thing that I could guide you to do is to calculate the fundamental group (or a covering space, or ...) of some of the spaces studied in the first half of the course. Concerning group theory, again I do not intend to ask you pure group theory problems on their own.
List of book sections. First half of the course: same as for MidTerm, but I will remind of things which I feel are sufficiently obscure.
Second half of the course: 51-55, 57-60, 67-81. (Less focus on the group theory sections.)

Information about the MidTerm:
1. Sections of the book covered: Sections 12-24, 26-28, 30-34, 43, 45-47 + statement of Tychonoff's Theorem.
2. Exam is closed book; you may quote any theorem in the main text of the book as long as you state it precisely in a form reasonably similar to how it is presented in the book, or in class.
3. The exam will probably have: some True-False questions for which no justification is required; some short questions asking for an example, or testing your knowledge of definitions, or asking you to apply an important result; several longer questions where a formal proof is required.
4. Here are some things I am not expecting you to know: definition of Lindelof, definition of completely regular, definition of locally compact, the examples involving S_\Omega, Theorems 33.2, 34.2, 34.3 (which I didnt cover in class).

Yes, the grades in the MidTerm will be scaled.

MidTerm Grading Blog
2pm: Started grading
2:10pm: Who didn't put their name on the exam? Please email me!
2:13pm: 18 papers in total. Average for TF questions: 5.3/10.
3pm: finished grading Questions 2-3.
3:10pm: Fortunately, the later questions are much more sparsely answered. I may yet make it to trick-or-treating.
3:20pm: There were many complete solutions to Q5, some solutions to Q6, but so far no one can do 4(b).
4:09pm: I'm done!!!!!!!
4:10pm: Highscore 93, shared by two students.
4:28pm: Average is 68. Median is 64.
4:40pm: I'll hand exams back on Monday. Happy Halloween!

Solution to 4(b). It is enough to check that if x(n) --> x in \Omega then g(x(n)) --> g(x). This is because \Omega, being a subspace of R^\omega, is metrizable. Since \sum_i x_i <= 1, we may find k so that x_k < epsilon/2. We may pick n so large that |x(n)_i - x_i| < epsilon/(2k) for all 1 <= i <= k. Then

(*) |\sum_{i=1}^k x_i^2 - \sum_{i=1}^k x(n)_i^2| < epsilon/2.

Also, x(n)_k, x(n)_{k+1},... are all < epsilon. But

(**) x(n)^2_k + x(n)^2_{k+1} +... < epsilon(x(n)_k+x(n)_{k+1}...+) < epsilon.


(***) x^2_k + x^2_{k+1} + ... < epsilon,

so combining (*),(**),(***), we get |g(x) - g(x(n))| < 3epsilon. This shows that g(x(n)) converges to g(x).